Problem 57 - Calculus: One and Several Variables

Question

$$\text { Set } f(x)=\frac{\sqrt{x^{2}-9}}{x^{2}}, x \geq 3 . \text { Use a CAS }$$ (a) 10 draw the graph of \(f\) (b) to find the area between the graph of \(f\) and the \(x\) -axis from \(x=3\) to \(x=6\) (c) to locate the centroid of the region described in part (b).

Step-by-Step Solution

Answer

The graph of the function \(f(x)\) is first drawn. Then, the area under this curve and above the x-axis from \(x = 3\) to \(x = 6\) is calculated via integration. Finally, the centroid of this area is found using the area calculated, and the formulas for the centroid's x and y coordinates.

Step 1: Graphing the function

In order to plot the function \(f(x)=\frac{\sqrt{x^{2}-9}}{x^{2}}\), input the function into the CAS (Computer Algebra System). The domain of the function is \(x \geq 3\). Thus, the x-value of the graph should be limited to values from 3 onward.

Step 2: Computing the area

The area between the curve of \(f(x)\) and the x-axis from \(x = 3\) to \(x = 6\) can be computed using integral calculus. Integrate the function \(f(x)\) from 3 to 6.

Step 3: Finding the centroid

The centroid \((\bar{x}, \bar{y})\) of an area under a curve can be found via the formulas \[\bar{x} = \frac{1}{Area} \int_{a}^{b} x f(x) dx\] and \[\bar{y} = \frac{1}{2*Area} \int_{a}^{b} [f(x)]^2 dx\]. In this case, the area is the one calculated in Step 2, and the limits of integration are \(x = 3\) and \(x = 6\). Substitute these values in the formulas to compute the coordinates of the centroid.

Key Concepts

Graphing Functions

Graphing functions is a powerful way to visually understand mathematical relationships and behaviors. For this exercise, the function given is \(f(x)=\frac{\sqrt{x^2-9}}{x^2}\). The domain restrictions specify that \(x\) should be greater than or equal to 3, meaning we only consider values from 3 onward. When plotting such a function, you use a CAS (Computer Algebra System) to accurately draw the graph. This tool will display the curve's shape, allowing you to see important features such as where it increases or decreases, which can be pivotal for solving calculus problems.
The graph helps you to anticipate where the function might intersect axes or approach asymptotes. Graphing is essential in calculus as it provides an intuitive grasp of the function’s behavior, helping you see how different values affect the output. Always remember, seeing a graph is like looking at a map — it’s easier to navigate and understand the lay of the land when you have a clear picture.

Integral Calculus

Integral calculus is about finding the total accumulation of a quantity, often represented as the area under a curve. For the problem at hand, you need to find the area between the graph of \(f(x)=\frac{\sqrt{x^2-9}}{x^2}\) and the \(x\)-axis from \(x=3\) to \(x=6\).
This is done by calculating a definite integral. The integral of \(f(x)\) from 3 to 6 gives you this area. In mathematical terms, it is expressed as \int_{3}^{6}f(x)dx\.
Performing this integral calculation in a CAS tool will give you the precise value of the area.
Integral calculus reveals the "total accumulation" aspect of functions, accounting for all changes along a defined interval. When interpreting graphs, the space between the curve and the axis represents tangible concepts, like distance, volume, or probability, in real-world problems.

Centroid Calculation

The centroid is the "center of mass" or the average position of all the points in a shape. To find the centroid in calculus, especially under a curve, you use specific formulas that involve definite integrals. Here, after finding the area between the curve \(f(x)=\frac{\sqrt{x^2-9}}{x^2}\) and the x-axis, we can calculate the centroid using two formulas:

• \(\bar{x} = \frac{1}{\text{Area}} \int_{3}^{6} x f(x) dx\): This gives the average x-coordinate, basically how far left or right the centroid is along the x-axis.
• \(\bar{y} = \frac{1}{2*\text{Area}} \int_{3}^{6} [f(x)]^2 dx\): This provides the average y-coordinate, indicating the height of the centroid above the x-axis.

These integrals calculate weighted averages, where the weight is the value of the function itself over the interval specified. By substituting the correct values and computed area into these formulas, you obtain the exact position of the centroid, offering a deeper insight into the balance point of the given region.